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Technologies for 3D data acquisition and 3D printing have enormously developed in the past few years, and, consequently, the demand for 3D virtual twins of the original scanned objects has increased. In this context, feature-aware denoising, hole filling and context-aware completion are three essential (but far from trivial) tasks. In this work, they are integrated within a geometric framework and realized through a unified variational model aiming at recovering triangulated surfaces from scanned, damaged and possibly incomplete noisy observations. The underlying non-convex optimization problem incorporates two regularisation terms: a discrete approximation of the Willmore energy forcing local sphericity and suited for the recovery of rounded features, and an approximation of the $$\ell _0$$ ℓ 0 pseudo-norm penalty favouring sparsity in the normal variation. The proposed numerical method solving the model is parameterization-free, avoids expensive implicit volume-based computations and based on the efficient use of the Alternating Direction Method of Multipliers. Experiments show how the proposed framework can provide a robust and elegant solution suited for accurate restorations even in the presence of severe random noise and large damaged areas.

Tubular and membranous shapes display a wide range of morphologies that are difficult to analyze within a common framework. By generalizing the classical Helfrich energy of biomembranes, we model them as solutions to a curvature optimization problem in which the principal curvatures may play asymmetric roles. We then give a novel phase-field formulation to approximate this geometric problem, and study its Gamma-limsup convergence. This results in an efficient GPU algorithm that we validate on well-known minimizers of the Willmore energy; the software for the implementation of our algorithm is freely available online. Exploring the space of parameters reveals that this comprehensive framework leads to a wide continuum of shape textures. This first step towards a unifying theory will have several implications, in biology for quantifying tubular shapes or designing bio-mimetic scaffolds, but also in computer graphics, materials science, or architecture.

By the classical Li–Yau inequality, an immersion of a closed surface in ℝ n {\mathbb{R}^{n}} with Willmore energy below 8 ⁢ π {8\pi} has to be embedded. We discuss analogous results for curves in ℝ 2 {\mathbb{R}^{2}} , involving Euler’s elastic energy and other possible curvature functionals. Additionally, we provide applications to associated gradient flows.

For a hypersurface in ${\mathbb R}^3$, Willmore flow is defined as the $L^2$--gradient flow of the classical Willmore energy: the integral of the squared mean curvature. This geometric evolution law is of interest in differential geometry, image reconstruction and mathematical biology. In this paper, we propose novel numerical approximations for the Willmore flow of axisymmetric hypersurfaces. For the semidiscrete continuous-in-time variants we prove a stability result. We consider both closed surfaces, and surfaces with a boundary. In the latter case, we carefully derive weak formulations of suitable boundary conditions. Furthermore, we consider many generalizations of the classical Willmore energy, particularly those that play a role in the study of biomembranes. In the generalized models we include spontaneous curvature and area difference elasticity (ADE) effects, Gaussian curvature and line energy contributions. Several numerical experiments demonstrate the efficiency and robustness of our developed numerical methods., 52 pages, 19 figures

We develop an approach to the conformal geometry of embedded hyper- surfaces by treating them as conformal infinities of conformally compact manifolds. This involves the Loewner-Nirenberg-type problem of finding on the interior a metric that is both conformally compact and of constant scalar curvature. Our first main result is an asymptotic solution to all orders. This involves log terms. We show that the coefficient of the first of these is a hypersurface conformal invariant which gener- alises to higher dimensions the Willmore invariant of embedded surfaces. We call this the obstruction density and for even dimensional hypersurfaces this is a fundamental curvature invariant. We make the latter notion precise and show that the obstruction density and the trace-free second fundamental form are, in a suitable sense, the only such invariants. We also show that this obstruction to smoothness is a scalar density analog of the Fefferman-Graham obstruction tensor for Poincare-Einstein metrics; in part this is achieved by exploiting Bernstein-Gel'fand-Gel'fand machinery. The solu- tion to the constant scalar curvature problem provides a smooth hypersurface defining density determined canonically by the embedding up to the order of the obstruction. We give two key applications: the construction of conformal hypersurface invariants and the construction of conformal differential operators. In particular we present an infinite family of conformal powers of the Laplacian determined canonically by the conformal embedding. In general these depend non-trivially on the embedding and, in contrast to Graham-Jennes-Mason-Sparling operators intrinsic to even dimensional hypersurfaces, exist to all orders. These extrinsic conformal Laplacian powers deter- mine an explicit holographic formula for the obstruction density. We also use these to construct conformally invariant hypersurface functionals. On each even dimensional hypersurface they provide a higher dimensional generalisation of the Willmore energy functional. In particular the functional gradient, with respect to variation of embed- ding, has linear leading term and thus provides another generalisation of the Willmore equation. It agrees with the obstruction density equation at leading derivative order.

In this paper we build an explicit example of a minimal bubble on a Willmore surface, showing there cannot be compactness for Willmore immersions of Willmore energy above $16 \pi$. Additionnally we prove an inequality on the second residue for limits sequences of Willmore immersions with simple minimal bubbles. Doing so, we exclude some gluing configurations and prove compactness for immersed Willmore tori of energy below $12 \pi$., Comment: 51 pages

Nanoscale materials have gained considerable interest because of their special properties and wide range of applications. Many types of boron nitride at the nanoscale have been realized, including nanotubes, nanocones, fullerenes, tori, and graphene sheets. The connection of these structures at the nanoscale leads to merged structures that have enhanced features and applications. Modeling the joining between nanostructures has been adopted by different methods. Namely, carbon nanostructures have been joined by minimizing the elastic energy in symmetric configurations. In other words, the only considerable curvature in the elastic energy is the axial curvature. Accordingly, because it has nanoscale structures similar to those in carbon, BN can also be joined and connected by using this method. On the other hand, different methods have been proposed to consider the rotational curvature because it has a similar size. Based on that argument, the Willmore energy, which depends on both curvatures, has been minimized to join carbon nanostructures. This energy is used to identify the joining region, especially for a three-dimensional structure. In this paper, we expand the use of Willmore energy to cover the joining of boron nitride nanostructures. Therefore, because catenoids are absolute minimizers of this energy, pieces of catenoids can be used to connect nanostructures. In particular, we joined boron nitride fullerene to three other BN nanostructures: nanotube, fullerene, and torus. For now, there are no experimental or simulation data for comparison with the theoretical connecting structures predicted by this study, which is some justification for the suggested simple model shown in this research. Ultimately, various nanoscale BN structures might be connected by considering the same method, which may be considered in future work.

The invariant theory for conformal hypersurfaces is studied by treating these as the conformal infinity of a conformally compact manifold. Recently it has been shown how, given a conformal hypersurface embedding, a distinguished ambient metric is found (within its conformal class) by solving a singular version of the Yamabe problem [21]. This enables a route to proliferating conformal hypersurface invariants. The aim of this work is to give a self contained and explicit treatment of the calculus and identities required to use this machinery in practice. In addition we show how to compute the solution’s asymptotics. We also develop the calculus for explicitly constructing the conformal hypersurface invariant differential operators discovered in [21] and in particular how to compute extrinsically coupled analogues of conformal Laplacian powers. Our methods also enable the study of integrated conformal hypersurface invariants and their functional variations. As a main application we prove that a class of energy functions proposed in a recent work have the right properties to be deemed higher-dimensional analogues of the Willmore energy. This complements recent progress on the existence and construction of different functionals in [22] and [20].

A conformally invariant generalization of the Willmore energy for compact immersed submanifolds of even dimension in a Riemannian manifold is derived and studied. The energy arises as the coefficient of the log term in the renormalized area expansion of a minimal submanifold in a Poincare-Einstein space with prescribed boundary at infinity. Its first variation is identified as the obstruction to smoothness of the minimal submanifold. The energy is explicitly identified for the case of submanifolds of dimension four. Variational properties of this four-dimensional energy are studied in detail when the background is a Euclidean space or a sphere, including identifications of critical embeddings, questions of boundedness above and below for various topologies, and second variation.

We develop an efficient calculus for varying hypersurface embeddings based on variations of hypersurface defining functions. This is used to show that the functional gradient of a new Willmore-like, conformal hypersurface energy agrees exactly with the obstruction to smoothly solving the singular Yamabe problem for conformally compact four-manifolds. We give explicit formulae for both the energy functional and the obstruction. Vanishing of the latter is a necessary condition for solving the vacuum cosmological Einstein equations in four spacetime dimensions with data prescribed on a conformal infinity, while the energy functional generalizes the scheme-independent contribution to entanglement entropy across surfaces to hypersurfaces.

This work investigates the formation of singularities under the steepest descent $$L^2$$ -gradient flow of $${{\,\mathrm{{ W}}\,}}_{\lambda _1, \lambda _2}$$ with zero spontaneous curvature, i.e., the sum of the Willmore energy, $$\lambda _1$$ times the area, and $$\lambda _2$$ times the signed volume of an immersed closed surface without boundary in $$\mathbb {R}^3$$ . We show that in the case that $$\lambda _1>1$$ and $$\lambda _2=0$$ , any immersion develops singularities in finite time under this flow. If $$\lambda _1 >0$$ and $$\lambda _2 > 0$$ , embedded closed surfaces with energy less than $$\begin{aligned} 8\pi +\min \left\{ \left( 16 \pi \lambda _1^3\right) \bigg /\left( 3\lambda _2^2\right) , 8\pi \right\} \end{aligned}$$ and positive volume evolve singularities in finite time. If in this case the initial surface is a topological sphere and the initial energy is less than $$8 \pi $$ , the flow shrinks to a round point in finite time. We furthermore discuss similar results for the case that $$\lambda _2$$ is negative. These results strengthen the ones of McCoy and Wheeler (Commun Anal Geom 24(4):843–886, 2016). For $$\lambda _1 >0$$ and $$\lambda _2 \ge 0$$ , they showed that embedded closed spheres with positive volume and energy close to $$4\pi $$ , i.e., close to the Willmore energy of a round sphere, converge to round points in finite time.

From Bäcklund to Darboux, this monograph presents a comprehensive journey through the transformation theory of constrained Willmore surfaces, a topic of great importance in modern differential geometry and, in particular, in the field of integrable systems in Riemannian geometry. The first book on this topic, it discusses in detail a spectral deformation, Bäcklund transformations and Darboux transformations, and proves that all these transformations preserve the existence of a conserved quantity, defining, in particular, transformations within the class of constant mean curvature surfaces in 3-dimensional space-forms, with, furthermore, preservation of both the space-form and the mean curvature, and bridging the gap between different approaches to the subject, classical and modern. Clearly written with extensive references, chapter introductions and self-contained accounts of the core topics, it is suitable for newcomers to the theory of constrained Wilmore surfaces. Many detailed computations and new results unavailable elsewhere in the literature make it also an appealing reference for experts.

Due to a variety of applications of nanoscaled materials, several researchers further investigate a joining between two nanostructures as a candidate for new potential applications. Here, the vertically joining between the nanotube with the nano-torus is investigated. Variational calculus is used to predict the joining curve between two nanostructures based on minimizing the elastic energy. Moreover, Willmore energy is also utilized to determine the join region especially for a three-dimensional structure. Since the surface of a catenoid is a minimizer obtained by the Willmore energy function, it is used to join two symmetric nanostructures. We find that these two approaches have less than 10% difference in the positions of the joining curves. These two methods might be used to design other hybrid nano-scaled structures.

The class of constrained Willmore (CW) surfaces in space-forms constitutes a Mobius invariant class of surfaces with strong links to the theory of integrable systems, with a spectral deformation [8], defined by the action of a loop of flat metric connections, and Backlund transformations [9], defined by a dressing action by simple factors. Constant mean curvature (CMC) surfaces in 3-dimensional space-forms are [25] examples of CW surfaces, characterized by the existence of some polynomial conserved quantity [21, 22, 24]. Both CW spectral deformation and CW Backlund transformation preserve [21, 22, 24] the existence of such a conserved quantity, defining, in particular, transformations within the class of CMC surfaces in 3-dimensional space-forms, with, furthermore [21, 22, 24], preservation of both the space-form and the mean curvature, in the latter case. A classical result by Thomsen [28] characterizes, on the other hand, isothermic Willmore surfaces in 3-space as minimal surfaces in some 3-dimensional space-form. CW transformation preserves [8, 9] the class of Willmore surfaces, as well as the isothermic condition, in the particular case of spectral deformation [8]. We define, in this way, a CW spectral deformation and CW Backlund transformations of minimal surfaces in 3-dimensional space-forms into new ones, with preservation of the space-form in the latter case. This paper is dedicated to a reader-friendly overview of the topic.

We study surface energies depending on the mean curvature in total spaces of Killing submersions, which extend the classical notion of Willmore energy. Based on a symmetry reduction procedure, we construct vertical tori critical for these mean curvature energies. These vertical tori are based on closed curves critical for curvature energy functionals in Riemannian 2-space forms. The binormal evolution of these critical curves in Riemannian 3-space forms generates rotational tori solutions for an ample family of Weingarten surfaces. Therefore, we also introduce some correspondence results between these two types of tori and illustrate their relation.

We consider a fluid interface in contact with an elastic membrane and study the static profiles of the interface and the membrane. Equilibrium conditions are derived by minimizing the total energy of the system with volume constraints. The total energy consists of surface energies and the Willmore energy; the latter penalizes the bending of the membrane. It is found that, while the membrane is locally flat at the contact line with the contact angle satisfying the Young-Dupré equation, the gradient of the mean curvature of the membrane exhibits a jump across the contact line. This jump balances the surface tension of the fluid interface in the normal direction of the membrane. Asymptotic solutions are obtained for two-dimensional systems in the limits as the reduced bending modulus ν tends to +∞ and 0, respectively. In the stiff limit as ν→+∞, the leading-order solution is given by that of a droplet sitting on a rigid substrate with the contact angle satisfying the Young-Dupré equation; in contrast, in the soft limit as ν→0, a transition layer appears near the contact line and the interfaces have constant curvatures in the outer region with apparent contact angles obeying Neumann's law. These solutions are validated by numerical experiments.

We discuss a variational model, given by a weighted sum of perimeter, bending and Riesz interaction energies, that could be considered as a toy model for charged elastic drops. The different contributions have competing preferences for strongly localized and maximally dispersed structures. We investigate the energy landscape in dependence of the size of the ‘charge’, that is, the weight of the Riesz interaction energy.In the two-dimensional case, we first prove that for simply connected sets of small elastica energy, the elastica deficit controls the isoperimetric deficit. Building on this result, we show that for small charge the only minimizers of the full variational model are either balls or centred annuli. We complement these statements by a non-existence result for large charge. In three dimensions, we prove area and diameter bounds for configurations with small Willmore energy and show that balls are the unique minimizers of our variational model for sufficiently small charge.

We demonstrate that Radon measures which arise as the limit of the Modica-Mortola measures associated to phase-fields with uniformly bounded diffuse area and Willmore energy may be singular at the boundary of a domain and discuss implications for practical applications. We furthermore give partial regularity results for the phase-fields $u_\epsilon$ at the boundary in terms of boundary conditions and counterexamples without boundary conditions., Comment: arXiv admin note: substantial text overlap with arXiv:1512.08641

We show that local minimizers of the Canham-Helfrich energy are asymptotically stable with respect to a model for relaxational fluid vesicle dynamics that we already studied in previous papers ([12, 11]). The proof is based on a Lojasiewicz-Simon inequality.

The main goal of this paper is to use the theory of exterior differential forms in deriving variations of the deformed Willmore energy in space forms and study the minimizers of the deformed Willmore energy in space forms. We derive both first and second order variations of deformed Willmore energy in space forms explicitly using moving frame method. We prove that the second order variation of deformed Willmore energy depends on the intrinsic Laplace Beltrami operator, the sectional curvature and some special operators along with mean and Gauss curvatures of the surface embedded in space forms, while the first order variation depends on the extrinsic Laplace Beltrami operator.

The conformal geometry of submanifolds in a constant curvature space was well studied in the past 15 years. The first part of this paper presents the system of complete conformal invariants of submanifolds in a general Riemannian space, and the second part presents several conformal rigidity theorems on compact Willmore hypersurfaces. In particular, the conformal class of Willmore tori is characterized using a conformal invariant.

In this paper, we will prove that any closed minimal Willmore hypersurface M 4 of S 5 with constant scalar curvature must be isoparametric. To be precise, M 4 is either an equatorial 4 sphere, a product of sphere S 2 ( 2 2 ) × S 2 ( 2 2 ) or a Cartan's minimal hypersurface. In particular, the value of the second fundamental form S can only be 0, 4, 12. This result strongly supports Chern's Conjecture.

We regularize non-convex anisotropic surface energy of a two-dimensional surface, given as a graph over the two-dimensional unit disk, by the Willmore functional and investigate existence of the corresponding global minimizers. Restricting to the rotationally symmetric case, we obtain a one-dimensional variational problem which permits to derive substantial qualitative information on the minimizers. We show that minimizers tend to a “cone”-like solution as the regularization parameter tends to zero. Areas where the solutions are either convex or concave are identified. It turns out that the structure of the chosen anisotropy hardly affects the qualitative shape of the minimizers.

For every $\;b>1\;$ fixed, we explicitly construct $1$-dimensional families of embedded constrained Willmore tori parametrized by their conformal class $\;(a,b)$\; with $\; a \sim_b 0^+\;$ deforming the homogenous torus \;$f^b$ of conformal class \;$(0,b).$ The variational vector field at $f^b$ is hereby given by a non-trivial zero direction of a penalized Willmore stability operator which we show to coincide with a double point of the corresponding spectral curve. Further, we characterize for $b \sim 1$, $b \neq 1$ and $a \sim_b 0^+$ the family obtained by opening the "smallest" double point on the spectral curve which is heuristically the direction with the smallest increase of Willmore energy at $f^b$. Indeed we show in \cite{HelNdi1} that these candidates minimize the Willmore energy in their respective conformal class for $b \sim 1$, $b \neq 1$ and $a \sim_b 0^+.$, 35 pages, originally a section of arXiv:1710.00533, we decided to split the paper due to length

Indexación Scopus For an embedded conformal hypersurface with boundary, we construct critical order local invariants and their canonically associated differential operators. These are obtained holographically in a construction that uses a singular Yamabe problem and a corresponding minimal hypersurface with boundary. They include an extrinsic Q-curvature for the boundary of the embedded conformal manifold and, for its interior, the Q-curvature and accompanying boundary transgression curvatures. This gives universal formulæ for extrinsic analogs of Branson Q-curvatures that simultaneously generalize the Willmore energy density, including the boundary transgression terms required for conformal invariance. It also gives extrinsic conformal Laplacian power type operators associated with all these curvatures. The construction also gives formulæ for the divergent terms and anomalies in the volume and hyper-area asymptotics determined by minimal hypersurfaces having boundary at the conformal infinity. A main feature is the development of a universal, distribution-based, boundary calculus for the treatment of these and related problems. © 2021 https://www-sciencedirect-com.recursosbiblioteca.unab.cl/science/article/pii/S0001870821001389?via%3Dihub

We consider a variational problem that consists of the Willmore energy of planar curves and a linear term in the curvature with a spatially heterogeneous coefficient. This coefficient is assumed to be a piecewise constant function. The variational problem arises from the modeling of the epidermal membrane in human skin. It has been predicted that the spatial heterogeneity of cell adhesion is one of the important factors in the protuberance formation of the membrane. We investigate the existence and stability of stationary points represented by symmetric graphs and find a condition for the existence of such stationary points by using the methods developed in the study of the Navier problem for Willmore energy. Especially, we give a necessary and sufficient condition for the existence of symmetric stationary points in terms of the gap size of the spatial heterogeneity. Moreover, we show some results on the stability of these stationary points. (c) 2021 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Superscript/Subscript Available

In the field of 3D head modeling and animation, feature points are often needed to mark important regions of the face and can be used to animate or deform the input model. However, an automatic detection of features remains a challenging task. This paper presents a novel approach to feature detection based on curvature and its derived descriptors, such as shape index, curvedness and Willmore energy. Four important feature regions are detected using the proposed approach - eyes, nose, mouth, ears. For each region, feature points are detected. Results show that the feature points are detected with sufficient accuracy for further use.

For every $g\in\mathbb{N}_0$ and $\epsilon>0$, we construct a smooth genus $g$ surface embedded into the unit ball with area $8\pi$ and Willmore energy smaller than $8\pi + \epsilon$. From this we deduce that a minimising sequence for Willmore's energy in the class of genus $g$ surfaces embedded in the unit ball with area $8\pi$ converges to a doubly covered sphere for all $g\in\mathbb{N}_0$. We obtain the same result for certain Canham-Helfrich energies with $\chi_K\leq 0$ without genus constraint and show that Canham-Helfrich energies with $\chi_K>0$ are not bounded from below in the class of smooth surfaces with area $S$ embedded into a domain $\Omega\Subset \mathbb{R}^3$. Furthermore, we prove that the class of connected surfaces embedded in a domain $\Omega\Subset\mathbb{R}^3$ with uniformly bounded Willmore energy and area is compact under varifold convergence.

We consider the numerical approximation of geometric Partial Differential Equations (PDEs) defined on surfaces in the 3D space. In particular, we focus on the geometric PDEs deriving from the minimization of an energy functional by L 2 -gradient flow. We analyze two energy functionals: the area, which leads to the mean curvature flow, a nonlinear second order PDE, and the Willmore energy, leading to the Willmore flow, a nonlinear fourth order PDE. We consider surfaces represented by single-patch tensor product NURBS and discretize the PDEs by means of NURBS-based Isogeometric Analysis in the framework of the Galerkin method. To approximate the high order geometric PDEs we use high order continuous NURBS basis functions. For the time discretization of the nonlinear geometric PDEs, we use Backward Differentiation Formulas (BDF) with extrapolation of the geometric quantities involved in the weak formulation of the problem; in this manner, we solve a linear problem at each time step. We report numerical results concerning the mean curvature and Willmore flows on different geometries of interest and we show the accuracy and efficiency of the proposed approximation scheme.

Using classical methods and solving certain differential equations we classify a kind of new surface, affine translation surfaces, which has the non-zero constant mean curvature in three dimensional Euclidean space $${{\bf E}^3}$$ . Therefore a kind of solutions for the constant mean curvature equation is given.

We conjecture that a Willmore torus having Willmore functional between 2π 2 and 2π 2 $$\sqrt 3 $$ is either conformally equivalent to the Clifford torus, or conformally equivalent to the Ejiri torus. Ejiri’s torus in S 5 is the first example of Willmore surface which is not conformally equivalent to any minimal surface in any real space form. Li and Vrancken classified all Willmore surfaces of tensor product in S n by reducing them into elastic curves in S 3, and the Ejiri torus appeared as a special example. In this paper, we first prove that among all Willmore tori of tensor product, the Willmore functional of the Ejiri torus in S 5 attains the minimum 2π 2 $$\sqrt 3 $$ , which indicates our conjecture holds true for Willmore surfaces of tensor product. Then we show that all Willmore tori of tensor product are unstable when the co-dimension is big enough. We also show that the Ejiri torus is unstable even in S 5. Moreover, similar to Li and Vrancken, we classify all constrained Willmore surfaces of tensor product by reducing them with elastic curves in S 3. All constrained Willmore tori obtained this way are also shown to be unstable when the co-dimension is big enough.

Noether’s theorem and the invariances of the Willmore functional are used to derive conservation laws that are satisfied by the critical points of the Willmore energy subject to generic constraints. We recover in particular previous results independently obtained by R. Capovilla and J. Guven, and by T. Rivière. Several examples are considered in detail.

Curvature analysis is important to the fingerprint image which is constructed by a group of regular ridgelines. Five kinds of curvatures constructed by the eigenvalues are tested, including the Gaussian curvature, Mean curvature, Laplacian curvature, Hessian curvature and our improved Hessian curvature. Researching on these five kinds of curvature, we find the relative advantages of our improved Hessian curvature for fingerprint curves. A measure for fingerprint reference point determination based on the eigenvalues of the covariance matrix of the ridges. Fingerprint reference points are determined by combining the global and local proposed curvature properties of ridges by using multi-resolution levels. The performance of the proposed one is verified through simulations and its analysis.

We use the minimizing movement theory to study the gradient flow associated to a non-regular relaxation of a geometric functional derived from the Willmore energy. Thanks to the coarea formula, we can define a Willmore energy on regular functions of L 1 L^{1} ( ℝ d \mathbb{R}^{d} ). This functional is extended to every L 1 {L^{1}} function by taking its lower semicontinuous envelope. We study the flow generated by this relaxed energy for radially non-increasing functions (functions with balls as superlevel sets). In the first part of the paper, we prove a coarea formula for the relaxed energy of such functions. Then, we show that the flow consists of an erosion of the initial data. The erosion speed is given by a first order ordinary equation.

This work considers a particular type of swept surface named canal surfaces in Euclidean 3-space. For such a kind of surfaces, some interesting and important relations about the Gaussian curvature, the mean curvature and the second Gaussian curvature are found. Based on these relations, some canal surfaces are characterized.

Let M 2 be a compact Willmore surface in the (2 + p)-dimensional unit sphere S 2 + p . Denote by H and S the mean curvature and the squared length of the second fundamental form of M 2, respectively. Set ρ 2 = S−2H 2. In this note, we proved that there exists a universal positive constant C, such that if ∥ρ 2∥2

We will show a symmetry result concerning Willmore surfaces of revolution which satisfy symmetric Dirichlet data. The first step is to find a regular energy minimizing surface. We will do this by carefully modifying a minimizing sequence using an idea by Dall’Acqua, Deckelnick & Grunau. After this we will establish a priori estimates for non-symmetric solutions, in which an order-reduction argument by Langer & Singer will be essential.

We propose a model for finding one-dimensional structure in a given measure. Our approach is based on minimizing an objective functional which combines the average-distance functional to measure the quality of the approximation and penalizes the curvature, similarly to the elastica functional. Introducing the curvature penalization overcomes some of the shortcomings of the average-distance functional, in particular the lack of regularity of minimizers. We establish existence, uniqueness and regularity of minimizers of the proposed functional. In particular we establish C1,1 estimates on the minimizers.

For a given family of smooth closed curves $\gamma^1,...,\gamma^\alpha\subset\mathbb{R}^3$ we consider the problem of finding an elastic \emph{connected} compact surface $M$ with boundary $\gamma=\gamma^1\cup...\cup\gamma^\alpha$. This is realized by minimizing the Willmore energy $\mathcal{W}$ on a suitable class of competitors. While the direct minimization of the Area functional may lead to limits that are disconnected, we prove that, if the infimum of the problem is $

In this paper, we study the critical case of the Allard regularity theorem. Combining with Reifenberg's topological disk theorem, we get a critical Allard-Reifenberg type regularity theorem. As a main result, we get the topological finiteness for a class of properly immersed surfaces in $\mathbb{R}^n$ with finite Willmore energy. Especially, we prove a removability of singularity of multiplicity one surface with finite Willmore energy and a uniqueness theorem of the catenoid under no a priori topological finiteness assumption., Comment: 38 pages

We present in this chapter fast operator-splitting-based algorithms for the solutions of variational problems from image processing. The models we consider use geometrical information and rely on the minimization of appropriate energy functionals. These energy functionals are nonquadratic, possibly nonconvex and nonsmooth, making their minimization a nontrivial endeavour. We show in this chapter that operator-splitting methods based on the Lie scheme, and variants of it, can provide algorithms that are efficient, robust and nearly parameter free. These new algorithms are simpler to use, and often more efficient, than those relying on alternating direction methods of multipliers (ADMM). We want to emphasize that one can use the methods discussed in this chapter for a wide range of applications; actually it has already been done for several of them.

The goal of the present paper is to investigate the algebraic structure of global conformal invariants of submanifolds. These are defined to be conformally invariant integrals of geometric scalars of the tangent and normal bundle. A famous example of a global conformal invariant is the Willmore energy of a surface. In codimension one we classify such invariants, showing that under a structural hypothesis (more precisely we assume the integrand depends separately on the intrinsic and extrinsic curvatures, and not on their derivatives) the integrand can only consist of an intrinsic scalar conformal invariant, an extrinsic scalar conformal invariant and the Chern-Gauss-Bonnet integrand. In particular, for codimension one surfaces, we show that the Willmore energy is the unique global conformal invariant, up to the addition of a topological term (the Gauss curvature, giving the Euler Characteristic by the Gauss Bonnet Theorem). A similar statement holds also for codimension two surfaces, once taking into account an additional topological term given by the Chern-Gauss-Bonnet integrand of the normal bundle. We also discuss existence and properties of natural higher dimensional (and codimensional) generalizations of the Willmore energy., 28 pages, final version to appear in the Annales de l'Institut Fourier

Formation of membrane necks is crucial for fission and fusion in lipid bilayers. In this work, we seek to answer the following fundamental question: what is the relationship between protein-induced spontaneous mean curvature and the Gaussian curvature at a membrane neck? Using an augmented Helfrich model for lipid bilayers to include membrane-protein interaction, we solve the shape equation on catenoids to find the field of spontaneous curvature that satisfies mechanical equilibrium of membrane necks. In this case, the shape equation reduces to a variable coefficient Helmholtz equation for spontaneous curvature, where the source term is proportional to the Gaussian curvature. We show how this latter quantity is responsible for non-uniform distribution of spontaneous curvature in minimal surfaces. We then explore the energetics of catenoids with different spontaneous curvature boundary conditions and geometric asymmetries to show how heterogeneities in spontaneous curvature distribution can couple with Gaussian curvature to result in membrane necks of different geometries., Comment: 17 pages, 10 figures

Curvature and curvature continuity, as two of the main geometry characteristics of complex surface, cause significant influence to the surface quality in the five-axis flank milling. Firstly, the curvature continuity grade in different complex surfaces have been defined accurately by introducing the basic concept about curvature and curvature continuity. Secondly, the relationship between curvature, curvature continuity and the machine tools movement are analyzed theoretically, it can be gotten that different curvature and continuity may lead to different machine tools movement law. Lastly, simulation results verify the correction about theoretical analysis, which is coincident with the theoretical analysis. The mechanism between curvature, curvature continuity and machine tools movement, contributes to achieve the optimization and improvement of complex surface quality.

In this paper we study the steepest descent $L^2$-gradient flow of the functional $\SW_{\lambda_1,\lambda_2}$, which is the the sum of the Willmore energy, $\lambda_1$-weighted surface area, and $\lambda_2$-weighted enclosed volume, for surfaces immersed in $\R^3$. This coincides with the Helfrich functional with zero `spontaneous curvature'. Our first results are a concentration-compactness alternative and interior estimates for the flow. For initial data with small energy, we prove preservation of embeddedness, and by directly estimating the Euler-Lagrange operator from below in $L^2$ we obtain that the maximal time of existence is finite. Combining this result with the analysis of a suitable blowup allows us to show that for such initial data the flow contracts to a round point in finite time., Comment: 31 pages

© 2016 Society for Industrial and Applied Mathematics.A new stable continuous-in-time semidiscrete parametric finite element method for Willmore flow is introduced. The approach allows for spontaneous curvature and area difference elasticity effects, which are important for many applications, in particular, in the context of membranes. The method extends ideas from Dziuk and the present authors to obtain an approximation that allows for a tangential redistribution of mesh points, which typically leads to better mesh properties. Moreover, we consider volume and surface area preserving variants of these schemes and, in particular, we obtain stable approximations of Helfrich flow. We also discuss fully discrete variants and present several numerical computations.